Her problem is the same as the first student's, except she knows what the first student did. Suppose she also sees a red marble.
If each marble is equally likely, the probability that a red marble is observed is 2/3 = P[B|A].
We want P[A|B], however. We can get it from Bayes' Rule, but we need to know P[A] and P[B].
The second student knows Bayes' Rule, and if she assumes the first student knows it and used it, she will make a different assumption about P[A] and P[B]:
P[A] = 2/3, P[B] = 1/2.
Then:
P[A|B] = P[B|A] (P[A] / P[B]) = (2/3) ((2/3)/(1/2)) = 8/9
and the student who sees a red marble should bet on "red".
Common mistake: substituting P[A|B] for P[B|A] without correcting for (possible) difference between P[A] and P[B].